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# Frontiers in Applied Mathematicsand Statistics

## Original Research ARTICLE Provisionally accepted The full-text will be published soon. Notify me

Front. Appl. Math. Stat. | doi: 10.3389/fams.2019.00047

# Efficient implementation of an iterative ensemble smoother for big-data assimilation and reservoir history matching

• 1NORCE Norwegian Research Centre, Norway
• 2Nansen Environmental and Remote Sensing Center, Norway
• 3Other, Norway

Raanes et al. [1] revised the iterative ensemble smoother of Chen and Oliver [2, 3], denoted
Ensemble Randomized Maximum Likelihood (EnRML), using the property that the EnRML
solution is contained in the ensemble subspace. They analyzed EnRML and demonstrated
how to implement the method without the use of expensive and potentially unstable pseudo
inversions of the low-rank state covariance matrix or the ensemble-anomaly matrix. The new
algorithm produces the same result, realization by realization, as the original EnRML method.
However, the new formulation is simpler to implement, numerically stable, and computationally
more efficient. The purpose of this document is to present a simple derivation of the new
algorithm and demonstrate its practical implementation and use for reservoir history matching.
An additional focus is to customize the algorithm to be suitable for big-data assimilation of
measurements with correlated errors. We demonstrate that the computational cost of the
resulting “ensemble sub-space” algorithm is linear in the number of measurements, also when
the measurements have correlated errors, as well as the state-space dimension. The final
algorithm is implemented in the Ensemble Reservoir Tool (ERT) for running and conditioning
ensembles of reservoir models. Several verification experiments are presented.

Keywords: EnRML, Iterative ensemble kalman smoother, History matching, Data assimilaiton, inverse methods, parameter estimation

Received: 01 Jul 2019; Accepted: 10 Sep 2019.

Copyright: © 2019 Evensen, Raanes, Stordal and Hove. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

* Correspondence: Dr. Geir Evensen, NORCE Norwegian Research Centre, Bergen, Norway, geir.evensen@gmail.com